Defining "Lucky"

 

What are the odds of Bob being randomly flipped into the air and landing on his head?

First, we have to define "landing on their head". Just because Bob's  head is the first body part to impact the ground does not mean brain-damage will result, especially if Bob is lucky enough to land face first. God pack a lot of soft, spongy energy absorbers in the front of your head: nose, sinuses, jaw, tongue and so on.

For the sake of this back-of-envelop calculation, let's assume the zone where there is a very high probability of brain or spinal injury is when Bob lands within 45 degrees of perfectly upside-down as shown in the picture above.

The odds of that happening are ((1.0 - cos theta))/2 where theta is the half-angle of the cone.

Pushing the numbers through the formula yields a result of 18% and since our starting assumptions (45 degree angle) were pulled out of thin air, mathematical integrity demands that we round to 20% to not imply more precision than is factual.

The odds if you use a 60 degree half-angle are 25% and are 5% for a 30 degree half-angle.

The first order approximation of Bob never again asking "Can we do it?" is 20% in the event he is randomly dropped dropped to the ground with enough energy to snap his neck.

Testing

This could be tested by taking fat, dry-erase markers and trimming the cap to expose an appropriate amount of tip.

Then, students could be instructed to randomly toss the marker at a pad positioned on the ground. Every student gets 20 tosses. Results are pooled.

20 students times 20 "simulations" per student is 400 runs. Count the number of ink-marks on the pad and divide by 400 and you will have another estimate of probability of very-bad outcomes.

Bonus link

Adventures with the Lower Level

The blog of an educator who tries to teach students practical problem solving methods so they put all that education to practical use.

This post was partially inspired by her(?) blog